Layerwise Terminal Discrepancy in Chen's Reverse-Heat Coupling on the Boolean Cube
This provides an incremental improvement to a known bound in Boolean analysis, relevant for researchers working on isoperimetric inequalities and convolution conjectures.
The paper refines Chen's reverse-heat coupling approach to Talagrand's convolution conjecture on the Boolean cube, achieving a \((\log\log\eta)^{1/2}\) improvement over Chen's result for the Boolean heat semigroup bound.
We isolate a layerwise refinement of the terminal testing-discrepancy step in Chen's perturbed reverse-heat approach~\cite{Chen2026} to Talagrand's convolution conjecture on the Boolean cube. Built on the joint-filtration martingale formulation of Chen's coupling, and on Chen's approximate monotonicity and conditional squared-score estimates being available in the joint-filtration form stated below, we prove the localized testing estimate \[ D_E\le C_τ\bigl(\cS_E+\sqrt{\cS_E\,\Pp(E)}\bigr), \qquad E\in\mathcal F_θ, \] where \(D_E\) is the localized terminal testing discrepancy and \(\cS_E\) is the stopped perturbative score energy. Applying this estimate to the layers \(G_r(θ)=\{r\le R_θ<r+1\}\) replaces the global Cauchy--Schwarz discrepancy cost by the layerwise cost \[ O_τ\left(\fracα{\sqrt r}+\frac{α^2}{r}\right) \Pp(G_r(θ)), \qquad α\simeq\log\logη. \] Under these imported joint-filtration inputs, combining the localized estimate with the time-smoothed anti-concentration profile yields the black-box consequence \[ μ\{P_τf>η\|f\|_1\} \le C_τ\frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3, \] for the Boolean heat semigroup. This makes a $(\log\logη)^{1/2}$ improvement over Chen's result.